Visual simultaneous localization and mapping (V-SLAM) is a fundamental capability for autonomous perception and navigation. However, endoscopic scenes violate the rigidity assumption due to persistent soft-tissue deformations, creating a strong coupling ambiguity between camera ego-motion and intrinsic deformation. Although recent monocular non-rigid SLAM methods have made notable progress, they often lack effective decoupling mechanisms and rely on sparse or low-fidelity scene representations, which leads to tracking drift and limited reconstruction quality. To address these limitations, we propose NRGS-SLAM, a monocular non-rigid SLAM system for endoscopy based on 3D Gaussian Splatting. To resolve the coupling ambiguity, we introduce a deformation-aware 3D Gaussian map that augments each Gaussian primitive with a learnable deformation probability, optimized via a Bayesian self-supervision strategy without requiring external non-rigidity labels. Building on this representation, we design a deformable tracking module that performs robust coarse-to-fine pose estimation by prioritizing low-deformation regions, followed by efficient per-frame deformation updates. A carefully designed deformable mapping module progressively expands and refines the map, balancing representational capacity and computational efficiency. In addition, a unified robust geometric loss incorporates external geometric priors to mitigate the inherent ill-posedness of monocular non-rigid SLAM. Extensive experiments on multiple public endoscopic datasets demonstrate that NRGS-SLAM achieves more accurate camera pose estimation (up to 50\% reduction in RMSE) and higher-quality photo-realistic reconstructions than state-of-the-art methods. Comprehensive ablation studies further validate the effectiveness of our key design choices. Source code will be publicly available upon paper acceptance.